Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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PARAMETER DEPENDENCE OF SMOOTH STABLE MANIFOLDS

  • Barreira, Luis (Centro de Analise Matematica, Geometria e Sistemas Dinamicos Departamento de Matematica Instituto Superior Tecnico Universidade de Lisboa) ;
  • Valls, Claudia (Centro de Analise Matematica, Geometria e Sistemas Dinamicos Departamento de Matematica Instituto Superior Tecnico Universidade de Lisboa)
  • Received : 2018.07.03
  • Accepted : 2018.10.12
  • Published : 2019.05.01
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Abstract

We establish the existence of $C^1$ stable invariant manifolds for differential equations $u^{\prime}=A(t)u+f(t,u,{\lambda})$ obtained from sufficiently small $C^1$ perturbations of a nonuniform exponential dichotomy. Since any linear equation with nonzero Lyapunov exponents has a nonuniform exponential dichotomy, this is a very general assumption. We also establish the $C^1$ dependence of the stable manifolds on the parameter ${\lambda}$. We emphasize that our results are optimal, in the sense that the invariant manifolds are as regular as the vector field. We use the fiber contraction principle to establish the smoothness of the invariant manifolds. In addition, we can also consider linear perturbations, and thus our results can be readily applied to the robustness problem of nonuniform exponential dichotomies.

Keywords

Acknowledgement

Supported by : FCT

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